We deduce from the conductor formula, conjectured by S. Bloch, the relation predicted by P. Deligne between the total dimension of the vanishing cycles and the Milnor number of an isolated singularity. Thanks to S. Bloch's work, we can apply this result to relative curves; in the appendix, the case of an arbitrary proper singular locus is considered as well.
Nous déduisons de la formule du conducteur, conjecturée par S. Bloch, celle de P. Deligne exprimant, dans le cas d'une singularité isolée, la dimension totale des cycles évanescents en fonction du nombre de Milnor. En particulier, la formule de Deligne est établie en dimension relative un; en appendice, on généralise cet énoncé au cas d'un lieu singulier propre.
Classification: 14H10, 14D06, 14B07, 14D15
Keywords: isolated singularity, Milnor number, Euler characteristic, Swan conductor, compactification, formal scheme
@article{AIF_2003__53_6_1739_0,
author = {Orgogozo, Fabrice},
title = {Conjecture de Bloch et nombres de Milnor},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {53},
number = {6},
year = {2003},
pages = {1739-1754},
doi = {10.5802/aif.1991},
zbl = {1065.14005},
mrnumber = {2038779},
language = {fr},
url = {http://www.numdam.org/item/AIF_2003__53_6_1739_0}
}
Orgogozo, Fabrice. Conjecture de Bloch et nombres de Milnor. Annales de l'Institut Fourier, Volume 53 (2003) no. 6, pp. 1739-1754. doi : 10.5802/aif.1991. http://www.numdam.org/item/AIF_2003__53_6_1739_0/
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